Non-vanishing modulo p of central critical Rankin-Selberg L-values with anticyclotomic twists

We prove non-vanishing modulo p, for a prime $\mathrm{\ell }$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\mathrm{\ell }$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.